Problem solving means engaging in a task for which the solution method is not
known in advance. In order to find a solution, students must draw on their
knowledge, and through this process, they will often develop new mathematical
understandings. Solving problems is not only a goal of learning mathematics
but also a major means of doing so. Students should have frequent opportunities
to formulate, grapple with, and solve complex problems that require a
significant amount of effort and should then be encouraged to reflect
on their thinking.

By learning problem solving in mathematics, students should acquire ways of thinking,
habits of persistence and curiosity, and confidence in unfamiliar situations
that will serve them well outside the mathematics classroom. In everyday
life and in the workplace, being a good problem solver can lead to great
advantages.

Problem solving is an integral part of all mathematics learning, and so it should
not be an isolated part of the mathematics program. Problem solving in
mathematics should involve all the five content areas described in these
Standards. The contexts of the problems can vary from familiar experiences
involving students' lives or the school day to applications involving
the sciences or the world of work. Good problems will integrate multiple
topics and will involve significant mathematics.

Build new mathematical
knowledge through problem solving

How can problem solving help students learn mathematics? Good problems
give students the chance to solidify and extend what they know and, when
well chosen, can stimulate mathematics learning. With young children,
most mathematical concepts can be introduced through problems that come
from their worlds. For example, suppose second graders wanted to find
out whether there are more boys or girls in the four second-grade classes.
To solve this problem, they would need to learn how to gather information,
record data, and accurately add several numbers at a time. In the middle
grades, the concept of proportion might be introduced through an investigation
in which students are given recipes for punch that call for different
amounts of water and juice and are asked to determine which is "fruitier."
Since no two recipes yield the same amount of juice, this problem is difficult
for students who do not have an understanding of proportion. As various
ideas are tried, with good questioning and guidance by a teacher, students
eventually converge on using proportions. In high school, many areas of
the curriculum can be introduced through problems from mathematical or
applications contexts.

Problem solving can and should be used to help students develop fluency with
specific skills. For example, consider the following problem, which is
adapted from the Curriculum and Evaluation Standards for School Mathematics
(NCTM 1989, p. 24):

I have pennies, dimes, and nickels in my pocket.
If I take three coins out of my pocket, how much money could I have taken?

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52

Knowledge is needed to solve this problemknowledge of the value
of pennies, dimes, and nickels and also some understanding of addition.
» Working on this problem offers good practice in addition
skills. But the important mathematical goal of this problemhelping
students to think systematically about possibilities and to organize and
record their thinkingneed not wait until students can add fluently.

The teacher's role in choosing worthwhile problems and mathematical tasks is
crucial. By analyzing and adapting a problem, anticipating the mathematical
ideas that can be brought out by working on the problem, and anticipating
students' questions, teachers can decide if particular problems will help
to further their mathematical goals for the class. There are many, many
problems that are interesting and fun but that may not lead to the development
of the mathematical ideas that are important for a class at a particular
time. Choosing problems wisely, and using and adapting problems from instructional
materials, is a difficult part of teaching mathematics.

Solve problems that
arise in mathematics and in other contexts

People who see the world mathematically are said to have a "mathematical
disposition." Good problem solvers tend naturally to analyze situations
carefully in mathematical terms and to pose problems based on situations
they see. They first consider simple cases before trying something more
complicated, yet they will readily consider a more sophisticated analysis.
For example, a task for middle-grades students presents data about two
ambulance companies and asks which company is more reliable (Balanced
Assessment for the Mathematics Curriculum 1999a). A quick answer found
by looking at the average time customers had to wait for each company
turns out to be misleading. A more careful mathematical analysis involving
plotting response times versus time of day reveals a different solution.
In this task, a disposition to analyze more deeply leads to a more complete
understanding of the situation and a correct solution. Throughout the
grades, teachers can help build this disposition by asking questions that
help students find the mathematics in their worlds and experiences and
by encouraging students to persist with interesting but challenging problems.

Posing problems comes naturally to young children: I wonder how long it would
take to count to a million? How many soda cans would it take to fill the
school building? Teachers and parents can foster this inclination by helping
students make mathematical problems from their worlds. Teachers play an
important role in the development of students' problem-solving dispositions
by creating and maintaining classroom environments, from prekindergarten
on, in which students are encouraged to explore, take risks, share failures
and successes, and question one another. In such supportive environments,
students develop confidence in their abilities and a willingness to engage
in and explore problems, and they will be more likely to pose problems
and to persist with challenging problems.

Apply and adapt a
variety of appropriate strategies to solve problems

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53

Of the many descriptions of problem-solving strategies, some of the
best known can be found in the work of Pólya (1957). Frequently cited
» strategies include using diagrams, looking for patterns,
listing all possibilities, trying special values or cases, working backward,
guessing and checking, creating an equivalent problem, and creating a
simpler problem. An obvious question is, How should these strategies be
taught? Should they receive explicit attention, and how should they be
integrated with the mathematics curriculum? As with any other component
of the mathematical tool kit, strategies must receive instructional attention
if students are expected to learn them. In the lower grades, teachers
can help children express, categorize, and compare their strategies. Opportunities
to use strategies must be embedded naturally in the curriculum across
the content areas. By the time students reach the middle grades, they
should be skilled at recognizing when various strategies are appropriate
to use and should be capable of deciding when and how to use them. By
high school, students should have access to a wide range of strategies,
be able to decide which one to use, and be able to adapt and invent strategies.

Young children's earliest experiences with mathematics come through solving problems.
Different strategies are necessary as students experience a wider variety
of problems. Students must become aware of these strategies as the need
for them arises, and as they are modeled during classroom activities,
the teacher should encourage students to take note of them. For example,
after a student has shared a solution and how it was obtained, the teacher
may identify the strategy by saying, "It sounds like you made an organized
list to find the solution. Did anyone solve the problem a different way?"
This verbalization helps develop common language and representations and
helps other students understand what the first student was doing. Such
discussion also suggests that no strategy is learned once and for all;
strategies are learned over time, are applied in particular contexts,
and become more refined, elaborate, and flexible as they are used in increasingly
complex problem situations.

Monitor and reflect
on the process of mathematical problem solving

Effective problem solvers constantly monitor and adjust what they are
doing. They make sure they understand the problem. If a problem is written
down, they read it carefully; if it is told to them orally, they ask questions
until they understand it. Effective problem solvers plan frequently. They
periodically take stock of their progress to see whether they seem to
be on the right track. If they decide they are not making progress, they
stop to consider alternatives and do not hesitate to take a completely
different approach. Research (Garofalo and Lester 1985; Schoenfeld 1987)
indicates that students' problem-solving failures are often due not to
a lack of mathematical knowledge but to the ineffective use of what they
do know.

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54

Good problem solvers become aware of what they are doing and frequently
monitor, or self-assess, their progress or adjust their strategies as
they encounter and solve problems (Bransford et al. 1999). Such reflective
skills (called metacognition) are much more likely to develop
in a classroom environment that supports them. Teachers play an important
role in helping to enable the development of these reflective habits of
» mind by asking questions such as "Before we go on, are we
sure we understand this?" "What are our options?" "Do we have a plan?"
"Are we making progress or should we reconsider what we are doing?" "Why
do we think this is true?" Such questions help students get in the habit
of checking their understanding as they go along. This habit should begin
in the lowest grades. As teachers maintain an environment in which the
development of understanding is consistently monitored through reflection,
students are more likely to learn to take responsibility for reflecting
on their work and make the adjustments necessary when solving problems.